An short exact sequence of $\mathfrak{g}$ of which head and tail are in category $\mathcal{O}$.

Question

Let $\mathfrak{g}$ be a finite-dimensional, semisimple Lie algebra over $\mathbb{C}$. Let $$ 0\rightarrow X \rightarrow Y \rightarrow Z \rightarrow 0 $$ be a short exact sequence of $\mathfrak{g}$-modules.

$\bf My \text{ } question:$ If $X,Z$ are in the category $\mathcal{O}$. Is it true that $Y$ in the category $\mathcal{O}$ as well?

Thank you very much!

Answer 1

The category $\mathcal{O}$ is closed under quotienting, submodules and finite direct sums, but not under extensions. Hence $Y$ need not be in $\mathcal{O}$ in general (see here).